An Axiomatic Foundation for the Expected Shortfall

In the recent Basel Accords, the expected shortfall (ES) replaces the value-at-risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most popular risk measure in financial regulation. Although ES is-in addition to many other nice properties-a coherent risk measure, it does not yet have an axiomatic foundation. In this paper, we put forward four intuitive economic axioms for portfolio risk assessment-monotonicity, law invariance, prudence, and no reward for concentration-that uniquely characterize the family of ES. Therefore, the results developed herein provide the first economic foundation for using ES as a globally dominating regulatory risk measure, currently employed in Basel III/IV. Key to the main results, several novel notions such as tail events and risk concentration naturally arise, and we explore them in detail. As a most important feature, ES rewards portfolio diversification and penalizes risk concentration in a special and intuitive way, not shared by any other risk measure.